An injective martingale coupling

We give an injective martingale coupling; in particular, given measures μ and ν in convex order on R such that ν is continuous, we construct a martingale transport such that for each y in the support of the target law ν there is a unique x in a support of the initial law μ such that (some of) the mass at x is transported to y . Then π has disintegration π ( d x , d y ) = ν ( d y ) δ θ ( y ) ( d x ) for some function θ . More precisely we construct a martingale coupling π of the measures μ and ν such that there is a set Γ μ such that μ ( Γ μ ) = 1 and a disintegration ( π x ) x ∈ Γ μ of π of the form π ( d x , d y ) = π x ( d y ) μ ( d x ) such that, with Γ π x a support of π x , we have # { x ∈ Γ μ : y ∈ Γ π x } ∈ { 0 , 1 } for all y and { y : # { x ∈ Γ μ : y ∈ Γ π x } = 1 } = supp ( ν ) . Moreover, if μ is continuous we may take Γ π x = supp ( π x ) for each x . However, we cannot also insist that Γ μ = supp ( μ ) .